One nose but two nostrils: Learn to align with sparse connections between two olfactory cortices.

The integration of neural representations in the two hemispheres is an important problem in neuroscience. Recent experiments revealed that odor responses in cortical neurons driven by separate stimulation of the two nostrils are highly correlated. This bilateral alignment points to structured inter-hemispheric connections, but detailed mechanism remains unclear. Here, we hypothesized that continuous exposure to environmental odors shapes these projections and modeled it as online learning with local Hebbian rule. We found that Hebbian learning with sparse connections achieves bilateral alignment, exhibiting a linear trade-off between speed and accuracy. We identified an inverse scaling relationship between the number of cortical neurons and the inter-hemispheric projection density required for desired alignment accuracy, i.e., more cortical neurons allow sparser inter-hemispheric projections. We next compared the alignment performance of local Hebbian rule and the global stochastic-gradient-descent (SGD) learning for artificial neural networks. We found that although SGD leads to the same alignment accuracy with modestly sparser connectivity, the same inverse scaling relation holds. We showed that their similar performance originates from the fact that the update vectors of the two learning rules align significantly throughout the learning process. This insight may inspire efficient sparse local learning algorithms for more complex problems.